Inverse Robin Spectral Problem for the p-Laplace Operator

Abstract

We study an inverse Robin spectral problem for the $p$-Laplacian with mixed Dirichlet--Robin boundary conditions. The unknown coefficient lies on an inaccessible Robin portion, and the data are the principal eigenvalue and nonlinear conormal flux on an accessible Dirichlet patch. For a coating of thickness $\epsilon \rho$ and conductivity ${\epsilon }^{p-1}$, we prove convergence of the principal eigenvalue and strong $W^{1,p}$ convergence of the eigenfunctions to an effective Robin problem with coefficient $h={\rho }^{-(p-1)}$. The Rayleigh characterization gives a two-sided comparison inequality, ordered uniqueness and weighted $L^1$ stability, and Fréchet differentiability of the eigenvalue map for $1<p<\infty$. For nonordered coefficients and $p\ge 2$, the eigenvalue and accessible flux determine the coefficient under the stated critical-set connectivity and local boundary-regularity hypotheses. Differentiability of the principal eigenfunction branch is proved for $p=2$ and imposed as a structural hypothesis for $p>2$; together with a quantitative linearized estimate, it yields conditional local stability. On finite-dimensional classes, branch differentiability and linearized injectivity yield local Lipschitz stability (with branch differentiability automatic when $p=2$), while monotone one-parameter families are identifiable from the principal eigenvalue.